Example: Travelling Salesperson Problem Start with any complete - - PDF document

Local search algorithms

Chapter 4, Sections 3–4

Outline

♦ Hill-climbing ♦ Simulated annealing ♦ Genetic algorithms (briefly) ♦ Local search in continuous spaces (very briefly)

Iterative improvement algorithms

In many optimization problems, path is irrelevant; the goal state itself is the solution Then state space = set of “complete” configurations; find optimal configuration, e.g., TSP

• r, find configuration satisfying constraints, e.g., timetable

In such cases, can use iterative improvement algorithms; keep a single “current” state, try to improve it Constant space, suitable for online as well as offline search

Example: Travelling Salesperson Problem

Start with any complete tour, perform pairwise exchanges Variants of this approach get within 1% of optimal very quickly with thou- sands of cities

Example: n-queens

Put n queens on an n × n board with no two queens on the same row, column, or diagonal Move a queen to reduce number of conflicts

h = 5 h = 2 h = 0

Almost always solves n-queens problems almost instantaneously for very large n, e.g., n = 1million

Hill-climbing (or gradient ascent/descent)

“Like climbing Everest in thick fog with amnesia”

function Hill-Climbing(problem) returns a state that is a local maximum inputs: problem, a problem local variables: current, a node neighbor, a node current ← Make-Node(Initial-State[problem]) loop do neighbor ← a highest-valued successor of current if Value[neighbor] ≤ Value[current] then return State[current] current ← neighbor end

Hill-climbing contd.

Useful to consider state space landscape

current state

• bjective function

state space

global maximum local maximum "flat" local maximum shoulder

Random-restart hill climbing overcomes local maxima—trivially complete Random sideways moves escape from shoulders loop on flat maxima

Simulated annealing

Idea: escape local maxima by allowing some “bad” moves but gradually decrease their size and frequency

function Simulated-Annealing(problem, schedule) returns a solution state inputs: problem, a problem schedule, a mapping from time to “temperature” local variables: current, a node next, a node T, a “temperature” controlling prob. of downward steps current ← Make-Node(Initial-State[problem]) for t ← 1 to ∞ do T ← schedule[t] if T = 0 then return current next ← a randomly selected successor of current ∆E ← Value[next] – Value[current] if ∆E > 0 then current ← next else current ← next only with probability e∆ E/T

Properties of simulated annealing

At fixed “temperature” T, state occupation probability reaches Boltzman distribution p(x) = αe

E(x) kT

T decreased slowly enough = ⇒ always reach best state x∗ because e

E(x∗) kT /e E(x) kT = e E(x∗)−E(x) kT

≫ 1 for small T Is this necessarily an interesting guarantee?? Devised by Metropolis et al., 1953, for physical process modelling Widely used in VLSI layout, airline scheduling, etc.

Local beam search

Idea: keep k states instead of 1; choose top k of all their successors Not the same as k searches run in parallel! Searches that find good states recruit other searches to join them Problem: quite often, all k states end up on same local hill Idea: choose k successors randomly, biased towards good ones Observe the close analogy to natural selection!

Genetic algorithms

= stochastic local beam search + generate successors from pairs of states

32252124

Selection Cross−Over Mutation

24748552 32752411 24415124

24 23 20

32543213

11

29% 31% 26% 14%

32752411 24748552 32752411 24415124 32748552 24752411 32752124 24415411 24752411 32748152 24415417

Fitness Pairs

Genetic algorithms contd.

GAs require states encoded as strings (GPs use programs) Crossover helps iff substrings are meaningful components

+ =

GAs = evolution: e.g., real genes encode replication machinery!

Continuous state spaces

Suppose we want to site three airports in Romania: – 6-D state space defined by (x1, y2), (x2, y2), (x3, y3) – objective function f(x1, y2, x2, y2, x3, y3) = sum of squared distances from each city to nearest airport Discretization methods turn continuous space into discrete space, e.g., empirical gradient considers ±δ change in each coordinate Gradient methods compute ∇f =

∂x1 , ∂f ∂y1 , ∂f ∂x2 , ∂f ∂y2 , ∂f ∂x3 , ∂f ∂y3

to increase/reduce f, e.g., by x ← x + α∇f(x) Sometimes can solve for ∇f(x) = 0 exactly (e.g., with one city). Newton–Raphson (1664, 1690) iterates x ← x − H−1

f (x)∇f(x)

to solve ∇f(x) = 0, where Hij = ∂2f/∂xi∂xj